| L(s) = 1 | − 4·9-s − 4·11-s + 8·23-s − 2·25-s − 4·29-s − 20·37-s + 4·43-s + 4·53-s + 24·67-s + 24·71-s + 8·79-s + 7·81-s + 16·99-s − 8·107-s + 4·109-s − 24·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 1.20·11-s + 1.66·23-s − 2/5·25-s − 0.742·29-s − 3.28·37-s + 0.609·43-s + 0.549·53-s + 2.93·67-s + 2.84·71-s + 0.900·79-s + 7/9·81-s + 1.60·99-s − 0.773·107-s + 0.383·109-s − 2.25·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9484819256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9484819256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.777187617560067873481223660344, −8.510015737391016441751130816610, −8.042524361855270348276077393488, −7.996390870601821903459245115572, −7.31988268569079607383077387761, −7.03754578731282198030165533125, −6.58656412064968387452214377441, −6.38707651237430913544279386737, −5.47776322684104849845502864634, −5.47672341591620997389356854269, −5.15193245279753842520083598060, −4.95414257832187937683493415845, −4.00401407283564639225576083628, −3.73549943503345842008182943524, −3.25331568384605900256693060298, −2.85364676120141824160742269994, −2.29368907945488457294277048964, −2.01710438296605247614068955043, −1.08842611463841129113246625441, −0.32602621176531306770886831326,
0.32602621176531306770886831326, 1.08842611463841129113246625441, 2.01710438296605247614068955043, 2.29368907945488457294277048964, 2.85364676120141824160742269994, 3.25331568384605900256693060298, 3.73549943503345842008182943524, 4.00401407283564639225576083628, 4.95414257832187937683493415845, 5.15193245279753842520083598060, 5.47672341591620997389356854269, 5.47776322684104849845502864634, 6.38707651237430913544279386737, 6.58656412064968387452214377441, 7.03754578731282198030165533125, 7.31988268569079607383077387761, 7.996390870601821903459245115572, 8.042524361855270348276077393488, 8.510015737391016441751130816610, 8.777187617560067873481223660344
